Dynamic manipulation of mechanical resonators in the high amplitude regime through optical backaction Mahmood Bagheri, Menno Poot, Mo Li, Wolfram P. H. Pernice, Hong X. Tang Department of Electrical Engineering, Yale University, New Haven, CT 06511 Cavity optomechanics enables active manipulation of mechanical resonators through backaction cooling and amplification1,2. This ability to control mechanical motion with retarded optical forces has recently spurred a race towards realizing a mechanical resonator in its quantum ground state3,4,5,6,7,8,9. Here, instead of quenching optomechanical motion, we demonstrate high amplitude operation of nanomechanical resonators by utilizing a highly efficient phonon generation process. In this regime, the nanomechanical resonators gain sufficient energy from the optical field to overcome the large energy barrier of a double well potential, leading to nanomechanical slow-down and zero frequency singularity, as predicted by early theories10. Besides fundamental studies and interests in parametric amplification of small forces11, optomechanical backaction is also projected to open new windows for studying discrete mechanical states12,13 and to foster applications14,15,16,17. Here we realize a non-volatile mechanical memory element, in which bits are written and reset via optomechanical backaction by controlling the mechanical damping across the barrier. Our study casts a new perspective on the energy dynamics in coupled mechanical resonator – cavity systems and enables novel functional devices that utilize the principles of cavity optomechanics. Classical and quantum dynamics of nanomechanical systems promise new applications in nanotechnology18,19 and fundamental tests of quantum mechanics in mesoscopic objects2,9. Recent development of nanoscale electromechanical (NEMS) and optomechanical systems has enabled cooling of mechanical systems to their quantum ground state7,8, which brings the possibility of quantum information processing with mechanical devices20,21. On the other hand, for practical application at room temperature — such as signal processing22 and mass/force sensing23 — it is desirable to operate nanomechanical devices at high amplitudes. However, scaling a mechanical resonator to nanoscale dimensions reduces its dynamic range dramatically24: when a coherent drive of increasing strength is applied to the nanomechanical resonator, Duffing nonlinearities shift the resonance frequency away from the drive and the amplitude hardly increases. Therefore, reaching high amplitude in a NEMS using coherent driving is difficult. Here we show that by exploiting cavity optomechanics these limitations of traditionally operated NEMS resonators can be overcome. Coupling a mechanical resonator to an optical cavity leads to a variety of effects such as optical bistability25, the optical spring effect26, motion-induced transparency27,28,29, and damping and amplification of the resonator’s thermal motion. It is the latter effect that enables us to operate our devices in the high-amplitude regime, where the nanomechanical resonator can overcome the large energy barrier that separates its two stable positions. This enables our observation of the long-sought after zero-frequency anomaly in a nanomechanical system10. Furthermore, by utilizing real-time optical cooling, we controllably quench the oscillator into one of its potential-well minima from the high-amplitude state. This way, we demonstrate all-optical operation of a mechanical memory element which stores information in the two mechanically stable states of the resonators. Such memories, described by a canonical double-well potential with a large central energy barrier, require large vibration amplitudes to switch between discrete, mechanical states30. Since the barrier is sufficiently larger than the thermal energy, our mechanical memory is non-volatile and immune to electromagnetic perturbation, environmental fluctuations and leakage31, and presents itself as a viable 2 building block for the current effort in developing mechanical computing engines32,33,34,35. A rendering of the optomechanical system used in this study is shown in Fig. 1a. It consists of a nanometre-sized flexural resonator integrated in an optical race-track cavity, fabricated from commercially available silicon-on-insulator (SOI) wafers. The nanomechanical resonators have dimensions of 10 μm × 500 nm × 110 nm, an effective mass of meff = 1.0 pg, and a fundamental resonance frequency of Ωm/2π ∼ 8 MHz with mechanical damping rate of Γ0/2 π = 2.1 kHz. The race-track cavity has a free spectral range of 2 nm and a typical linewidth of 10 GHz. Optomechanical interactions are mediated by optical gradient forces between the nanomechanical resonator and the adjacent substrate36. The mechanical resonator is separated from the substrate by about 250 nm to have large optomechanical coupling rates (g/2π = 1 GHz/nm where g = dωc/dx, see Supp. Mat.) and yet to allow large oscillation amplitudes. Due to the residual compressive stress introduced by the SOI wafer bonding process, the free- standing doubly clamped beams are slightly buckled and have two stable configurations at rest37: buckled up and buckled down (see Fig. 1b). Therefore, the out-of-plane motion of the buckled beam can be described by a double-well potential, where both the ‘up’ and ‘down’ states correspond to the minima in the potential. The thermomechanical displacement noise spectra in Fig. 1e show that the two states have slightly different mechanical resonance frequencies, which indicates that the double-well potential is not completely symmetric. The two mechanical states are discriminated in optical transmission measurements because the optical mode has a different effective refractive index in the two states: when the waveguide is closer to the substrate (buckled-down state) the effective refractive index is larger than in the buckled-up state as the optical mode interacts stronger with the substrate. Therefore the optical cavity resonance shifts towards longer wavelengths when the resonator flips from the buckled-up state to the buckled-down state. Consequently, the optical cavity has distinct optical resonances in the two stable configurations as shown in low power optical transmission spectra in Fig. 1c. The low 3 power spectrum probes the static optomechanical resonances, but at high power the mechanical resonator starts to oscillate when the pump wavelength is scanned close to the optical resonance. Fig. 1d shows the optical transmission spectrum measured when the input optical power is well above the threshold for self-sustained oscillations (SSO) of ~600 μW. When the wavelength is scanned across the optical resonance, the transmission no longer shows the low-power Lorentzian shape, rather the resonance is dragged from the “up” to the “down” state: as soon the laser is blue detuned w.r.t. the “up” state, the self-sustained oscillations start and the cavity frequency oscillates back and forth with an amplitude Ap-p·g, where Ap-p is the mechanical resonator oscillation amplitude. The SSO in turn modulate the optical transmission at the mechanical oscillation frequency, indicated by the high-frequency components of the optical transmission (Fig. 1d). The SSO appear in the entire wavelength range between the “up” and “down” state optical modes, which indicates that the energy of the resonator exceeds the energy barrier between the two states. Stable self-sustained oscillators are highly desirable as they can provide low-noise high-frequency signals for potential applications in precision on-chip timekeeping, communication technology, and sensing38. Generally, to reduce the effect of noise in such applications, it is essential to increase the oscillation amplitude as much as possible by pumping a large number of phonons into the oscillator. The high amplitude optomechanical system we study here has very distinctive nonlinear dynamics in both the mechanical and optical domain. In the conventional framework of 1,2 cavity optomechanics (see Fig. 2a), a blue detuned input pump laser (at frequency ωp) resonantly enhances the Stokes sideband, while the off-resonant anti-Stokes sideband is suppressed. It is therefore more likely for a pump photon in the cavity to emit a phonon than to absorb one, thereby amplifying the motion. This dynamical backaction of the cavity on the resonator can be represented by a negative damping rate ΓBA which reduces the total mechanical damping rate Γ from its intrinsic value Γ0 to Γ0 + ΓBA. On the other hand, for a red-detuned pump, ΓBA is positive and leads to cooling. When the blue-detuned pump power is large enough ΓBA can become so negative that the total 4 damping rate of the mechanical resonator Γ0 + ΓBA vanishes, leading to regenerative oscillations sustained by the static pump power. In cavity optomechanics, the resolved sideband regime (RSR) is regarded as the most efficient way to cool the mechanical resonator. One might expect that the same is true for amplification. This is, however, untrue, since in the RSR only a single sideband is available, and one pump photon can only emit a single phonon39, as illustrated in Fig. 2b. The unresolved sideband regime (USR) is more preferable for efficient phonon generation. In this case many sidebands (Fig. 2c) are available for phonon emission and absorption and can even interfere with each other40. Consequently, the net effect is that a single blue-detuned photon can emit many phonons41. (A detailed analysis of phonon generation process